In the life cycle of a reservoir development, numerical reservoir simulations (hereinafter, “reservoir simulation”) become indispensable to understand the fluid flow and the distribution of underground fluid. In a simulation of a fluid flow in an oil reservoir, the equations governing fluid flow are solved by finite difference techniques. A fluid flow simulation typically uses a mesh of Cartesian grid cells (hereinafter, a “grid”) to represent an oil reservoir. To achieve the necessary numerical accuracy, the grid cells need to be very small in the vicinities of the well bore of the reservoir. In the following description, the grid cells adjacent to the well bore will be referred to as “near well bore cells”, and the cell(s) that contains the well bore will be referred to as the “well bore cell”.
In the conventional approaches of reservoir fluid flow simulation, there is a major constraint that governs the minimum dimensions of the near well bore cells and the well bore cell(s). The restraint can be described as follows. In conventional fluid flow simulations, a simulator determines the flowing completion pressure by the following equation:
                                          T            wj                    =                                    c              ⁢                                                          ⁢              θ              ⁢                                                          ⁢              Kh                                                      ln                ⁡                                  (                                                            R                      o                                                              R                      w                                                        )                                            +              s                                      ,                            (        1        )            where:
Twj denotes a connection transmissibility factor;
c denotes a unit conversion factor;
θ denotes the angle of the segment connecting with the cell;
Kh denotes the effective permeability times net thickness of the connection;
Ro denotes the “pressure equivalent radius” of the grid;
Rw denotes the well bore radius; and
s denotes the skin factor.
As shown in equation (1), the calculation of the connection transmissibility (i.e., connection transmissibility factor Twj) to the well bore involves a “pressure equivalent radius” variable R0, which is the distance from the well to where the local pressure is equal to the nodal average pressure of the grid. A pressure equivalent radius for a vertical well can be determined by the following equation:
                              R          0                =                  0.28          ⁢                                                                                          D                    x                    2                                    ⁢                                                                                    K                        y                                                                    K                        x                                                                                            +                                                      D                    y                    2                                    ⁢                                                                                    K                        x                                                                    K                        y                                                                                                                                                                                      K                    y                                                        K                    x                                                  4                            +                                                                    K                    x                                                        K                    y                                                  4                                                                        (        2        )            where R0 denotes pressure equivalent radius, Dx and Dy denote the x-direction and the y-direction dimensions of the grid cell, respectively, and Kx and Ky denote the x-direction and y-direction permeability, respectively. The following article described the above-referenced equation (2) and is incorporated herein by reference: Donald W. Peaceman, Interpretation of Well-Block Pressures in Numerical Reservoir Simulation With Nonsquare Grid Blocks and Anisotropic Permeability, SPE 10528, 1983.
The involvement of pressure equivalent radius R0 in equation (1) causes a constraint because conventional approaches cannot handle the situation where a pressure equivalent radius R0 is smaller than the well bore radius Rw. On the other hand, for reasons of numerical accuracy, it is undesirable to have a well bore cell much larger than the near well bore cells. That is, the well bore cell and the near well bore cells all need to be sufficiently small such that stiff numerical problems can be solved, which relates to the solution of the near well bore behavior. But as noted above, a pressure equivalent radius cannot be smaller than the well bore radius.